Sum and Product Rules
Exercise. Consider tossing a coin ve times. What is the probability of getting the same result on the rst two tosses or the last
two tosses?
Solution.
Let E be the event that the rst two tosses are the same and F
be the event that th

CSCA67Lecture2PermutationsandSelections
CountingPizzaToppings(DifferentMethod):
Q: commercial deal offered 2 pizzas, up to 5 toppings on each(nodoublingontoppings), 11
A
toppings to choose from. The commercial claimed there are 1,048,576 possibilities. Wh

University of Toronto at Scarborough
CSC A67/MAT A67 - Discrete Mathematics, Fall 2016
Exercise #3: Logic and Proofs
Due: September 28, 2016 at 11:59 p.m.
This exercise is worth 3% of your final grade.
Warning: Your electronic submission on MarkUs affirms

University of Toronto at Scarborough
CSCA67 / MATA67Discrete Mathematics, Fall 2016
Exercise #5: Induction / Basic Counting
Due: November 4, 2016 at 11:59 p.m.
This assignment is worth 3% of your final grade.
Warning: Your electronic submission on MarkUs

University of Toronto at Scarborough
CSC A67/MAT A67 - Discrete Mathematics, Fall 2016
Exercise #2: Logic and Proofs Solutions
Due: September 21, 2016 at 11:59 p.m.
This exercise is worth 3% of your final grade.
Warning: Your electronic submission on Mark

University of Toronto at Scarborough
CSC A67/MAT A67 - Discrete Mathematics, Fall 2016
Exercise #1: Implication
Due: September 14, extended to September 16 2016 at 11:59 p.m.
This exercise is worth 3% of your final grade.
Warning: Your electronic submissi

University of Toronto at Scarborough
CSC A67/MAT A67 - Discrete Mathematics, Fall 2016
Exercise #4: Indirect Proofs
Due: October 7, 2016 at 11:59 p.m.
This exercise is worth 3% of your final grade.
Warning: Your electronic submission on MarkUs affirms tha

History of Pascals Triangle
Pascal was not the rst to discover the triangle of binomial coefcients but was given credit because of how he related it to his
work with probability and expectation.
The triangle may have rst appeared more than 300 years earli

Counting Pizza Toppings Continued.
Q. How do we know that our calculation is the correct answer?
A. One way to convince ourselves is try to nd another way to
count the same problem.
We counted all possible orders and then removed duplicates.
Exercise. Rec

Counting With Repetitions
The genetic code of an organism stored in DNA molecules consist of 4 nucleotides:
Adenine, Cytosine, Guanine and Thymine.
It is possible to sequence short strings of molecules.
One way to sequence the nucleotides of a longer st

Trees
Defn. A tree is a connected graph without cycles or loops.
Trees are important data structures in computer science.
Youve already seen a rooted tree.
The Family Tree is a rooted tree. The top of the tree is the root
and the nodes follow a parent-chi

Boolean Logic
Another Scheduling Problem
Given a set of employees, schedule a meeting so that
everyone can attend. Assume the day is split up into time
slots and each person has a calendar which says whether
they are available for any given time slot.
Giv